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User talk:RSchlafly (view source)
Revision as of 13:38, February 16, 2009
, 13:38, February 16, 2009→Continuum: new section
{{cquote|'''Cheers to a new year and another chance for us to get it right!'''<small><small>--Oprah Winfrey</small></small>}}
{{cquote|'''Cheers to a new year and another chance for us to get it right!'''<small><small>--Oprah Winfrey</small></small>}}
--[[User:TK|'''₮K''']]<sup>[[User_Talk:TK|/Talk!]]</sup> 23:33, 31 December 2008 (EST)
--[[User:TK|'''₮K''']]<sup>[[User_Talk:TK|/Talk!]]</sup> 23:33, 31 December 2008 (EST)
== Continuum ==
Hi Roger. Foxtrot and I have been arguing over the Continuum article and we really need someone to step in and mediate. I have a problem with the entire second paragraph really: "The continuum is called so because it was the first (and most prominent) continuous set studied by mathematicians. No additional numbers may be added to the continuum (real line) without losing its dense linear order without endpoints. Since the complex numbers add the imaginary number, i, to the real line, this is one reason they have no natural linear order. Any linear order of the complex numbers needs the Axiom of Choice to be constructed."
First of all the reference to a continuous set is unneccessary. I had never heard of such a thing. When I googled it I saw that it is an obscure notion that has no place in such a short introductory article to the topic. I see no reason to believe that the continuum was called so because it is a continuous set. My best guess is that the use of the word continuum predates the use of the term continuous set.
The second sentence about not being able to add more numbers to the reals is essentially meaningless. What do we mean by number? We are discussing the real line as an order theoretic/topological object. It is easy to extend the reals as such an object with out losing the dense linear order. This ties into the last statement. I outline in http://www.conservapedia.com/index.php?title=Talk:Continuum&curid=81756&diff=621255&oldid=621024 how to construct the lexicographic order on the complex plane. This refutes the last sentence as such an ordering doesn't use AC. In addition the complex numbers ordered in such a way is, from an order theoretic point of view, <math> \mathfrak{c}</math> many copies of the reals laid out end-to-end (if you get my meaning). This shows that not only can we add a point to the reals we can add uncountably many copies of the reals to the reals without destroying the dense linear ordering.
Thanking you in advance for your time. [[User:AndyJM|AndyJM]] 08:38, 16 February 2009 (EST)